Questions tagged [projective-modules]

For questions about projective modules over a ring and projective objects in related categories.

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Hom functor and arbitrary coproducts

Let $M$ be a finitely generated $R$-module. It's easy to check that, in this case, $\mathbf{Hom}(M,-)$ preserves infinite sums. Now suppose that $M$ is projective. Is the reciprocal true? That is, if ...
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431 views

Basis for free modules over an affine domain

Let $A=k[x_1,\cdots,x_r]/I$ for some prime ideal $I$ and some field $k$. Consider the free $A$-module $A^n$. Question 1. Given an element $e\in A^n$, is there a method to tell whether $e$ can be ...
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Weyl theorem - possible corollary - alternative characterization of projective representation of $Z_N\times Z_N$

For an integer $N$, let $\omega=e^{2i\pi/N}$ and $A$, $B$ be the clock and shift operators: $A=\left(\begin{matrix} 1 & 0 & \cdots & 0 \\ 0 & \omega & \cdots & 0 \\ \vdots &...
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230 views

Property of the trace on finitely generated projective modules

Let $A$ be a commutative ring with unit and let $P$ be a projective $A$-module finitely generated. By definition, there exists an $A$-module $P'$ such that $P\oplus P'$ is free of finite rank $r$. If $...
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1answer
104 views

A weak Schur's lemma for non-semisimple finite dimensional algebras

Let $B \subseteq C$ be an inclusion of finite dimensional (associative) algebras over a field $k$. Assume that $C$ is a free $B$-module. Let $\bigoplus_i U_i$ be a decomposition of $B$ into ...
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54 views

Lifting idempotents and projective coverings — reference request

Let $R$ be a (non-commutative, unital) ring with Jacobson radical $J$, write $\overline{R}=R/J$ and denote the quotient map $R\to \overline{R}$ by $a\mapsto \overline{a}$. It is easy to see that if ...
2
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1answer
194 views

A module associated to an endomorphism of a vector bundle

Let $E$ be a vector bundle over a compact connected Hausdorff space $X$. To an endomorphism $\alpha \in End(E)$, we associate a $C(X)-$ module $\Gamma(E,\alpha)$ consisting of all $\beta\in End(E)$ ...
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On existence of finitely generated projective generator with commutative endomorphism ring in ${}_R Mod$

Let $R$ be a ring with unity (not necessarily commutative). Let ${}_R Mod$ be the category of left $R$-modules. If ${}_R Mod$ has a projective generator with commutative endomorphism ring , then does $...
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119 views

Projective modules over maximal orders of central simple algebras

In "Supersingular K3 surfaces" by TetsuJi Shioda, when proving Theorem 3.5 (Deligne) he considers a supersingular elliptic curve $C$ over an algebraic closed field of $\text{char}\ p>0$ and let $R ...
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1answer
429 views

Moduli space of flat connections over a Riemann surface

If I understand correctly, in the Refs below: We can see that the moduli space of SU($N$) flat connections over a torus, is equivalent to a complex projective space $\mathbb{P}^{N-1}$ Namely, $$ M_{\...
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158 views

Moduli space is a Calabi-Yau manifold?

I asked a question here where a moduli space of flat connection is related to the $n$-dimensional complex projective space: $$\Bbb E/S_n \cong \Bbb P^{n-1}. $$ This is related to a 4d SU(N) Yang-...
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1answer
50 views

Decomposing semihereditary rings

Let $R$ be a commutative semihereditary ring (i.e. every finitely generated ideal of $R$ is projective https://en.wikipedia.org/wiki/Hereditary_ring). Then is $R$ a finite direct product of Prufer ...
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1answer
82 views

Decomposing Noetherian hereditary rings of Krull dimension $1$ into product of hereditary domains (i.e. Dedekind domains)

Let $R$ be a commutative Noetherian hereditary ring (https://en.wikipedia.org/wiki/Hereditary_ring) of Krull dimension $1$. Then is it true that $R$ is a finite direct product of Dedekind domains ?
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33 views

Numerator-cancellable Modules

I don't know why there is no investigation of the cancellability of quotients in category of modules. What I mean by cancellability of quotients in category of modules is the following : Let $R$ be a ...
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1answer
193 views

Are these two constructions of $K_0(A)$ isomorphic?

The following question is extracted from this question on MSE, which got no answer so far, probably because it was a bit hidden by another question which a posteriori was totally obvious. Let $A$ be ...
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111 views

structure of flat modules over a DVR

I. Kaplansky has proved that if a torsion free module (over a complete DVR) is countably generated and does not contains infinitely divisible elements then it is free. Is there any analog of this ...
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1answer
392 views

A projective (or free) $\mathbb{Z}\pi_1$-module

Suppose that $Z$ is a finite wedge of spheres containing circles and there exist maps $f:Y\to Z$ and $g:Z\to Y$ so that $g\circ f\simeq 1_Y$. Assume that there exists a map $h:X\to Y$ which induces ...
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118 views

A question concerning a short exact sequence with an action

Let $A$ and $D$ be two non-trivial abelian groups and $B,C$ be two non-abelian groups. Also, let $C$ is a free group and acts on $A,B,D$. Let $0\to A \xrightarrow{f}B\xrightarrow{g}C\to 0$ be a ...
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1answer
198 views

Finitely generated submodule of non-finitely generated projective module is contained in some proper direct summand ?

Let $P$ be a non-finitely generated projective module over a commutative Noetherian ring. Is every finitely generated submodule of $P$ contained in some finitely generated direct summand of $P$ ? Or ...
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1answer
105 views

When can every countably generated submodule of a a non-countably generated projective module be contained in a countably generated direct summand ?

One answer to this Lemma on infinitely generated projective modules shows that every finitely generated module of a non-countably generated projective module is contained in a countably generated ...
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1answer
335 views

Adjoints of scalar extension and scalar coextension

Let $h\colon R\rightarrow S$ be a morphism of commutative rings. We consider the following functors (I am aware that the notations may be different in other contexts): $h^*$: Scalar extension by ...
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88 views

Finitely generated submodules of projectives lie inside f. g. projectives?

Let $R$ be a (not necessarily commutative) ring. If $M$ is a finitely generated submodule of a projective module $P$, is there a finitely generated projective submodule $P'$ such that $M \subseteq P'...
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1answer
233 views

If $M \otimes -$ is continuous, why is $M$ f.g. projective? Alternative proof

Let $R$ be a commutative ring and $M$ be some $R$-module such that $M \otimes -$ is continuous (i.e. preserves all limits). Then one can show that $M$ is f.g. projective. One way to prove this is to ...
7
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1answer
258 views

Role of stably free modules in algebraic geometry

For any ring $R$, a non-zero module $S$ is stably free if $S\oplus R^a$ is free ($a\geq 1$). This may be an overly vague question, but I am wondering in what contexts do stably free modules arise in ...
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1answer
204 views

Does $A\oplus M_n(R)\cong B\oplus M_n(R)$ imply $A\cong B$? $R$ Dedekind domain

Let $R$ be a Dedekind domain and $A, B$ be finitely generated projective $M_n(R)$-modules. Is it true that $A\oplus M_n(R)\cong B\oplus M_n(R)\:\:\Rightarrow\:\:A\cong B$? Here, the isomorphism is ...
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2answers
489 views

The projective covers of Artinian module

The injective hull for a module always exists, however over certain rings modules may not have projective covers. I have a question. If $A$ is an Artinian module on a Noetherian local ring $R$ then $...
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1answer
384 views

When is countable direct-product of projective modules again projective ?

Let $R$ be a commutative ring with unity. The Bass-Papp theorem states that any countable direct sum of injective $R$-modules is injective iff $R$ is Noetherian . Chase's theorem states that any ...
2
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1answer
285 views

On graded projective modules

If $R$ is a PID, then we know that any projective module over $R$ is free. Is there any graded version of this? That is, let us call a graded commutative ring $R=\oplus _{g \in G} R_g$ a graded PID if ...
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1answer
201 views

Why does there exist a non-split sequence with the condition that $\mathrm{pd} M=\infty$?

Why does there exist a non-split sequence with the condition that $\mathrm{pd} M = \infty$? Remarks. I am reading Andrzej Skowroński, Sverre O.Smalø, Dan Zacharia: On the Finiteness of the Global ...
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1answer
230 views

On integral domains over which special kind of modules are projective

For an integral domain $R$ let $\mathrm{Frac}(R)$ denote its field of fractions. Then $R$ is embedded in $\mathrm{Frac}(R)$ and we can consider $\mathrm{Frac}(R)$ as an $R$-module. Can we ...
2
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1answer
141 views

Is projectivity preserved by invariants?

Let $f:R\rightarrow S$ be a homomorphism of (commutative) rings with unity and suppose that $G$ is a group acting on $R$ and $S$ in such a way that $f\sigma=\sigma f$ for every $\sigma\in G$. Denote ...
7
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1answer
171 views

Graded and projective (but not bounded below) module that is not graded-projective?

Let $A = \bigoplus_{n = 0}^\infty A_n$ be a graded algebra over a field $k$ that is locally finite: each $A_n$ is a finite-dimensional $k$-vector space. We say that a graded left $A$-module $P = \...
8
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1answer
494 views

Kaplansky conjecture (consequences)

The Kaplansky conjecture says that: for any field $F$ and any torsion free group $G$, the group ring $F[G]$ does not have nontrivial idempotent elements. Questions Do we assume that $F$ has any ...
5
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3answers
347 views

Non-free projective pearls (general and Abelian)

A pearl is an ordered pair $\ \mathbf P:=(G\,\ S),\ $ where $\ G\ $ is a group, and $\ S\ $ is a non-empty subset of G which does not contain the neutral element of $\ G\ $ (i.e. not 1 in the ...
3
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2answers
146 views

Operations on semi-hereditary rings

I am learning about left (right) semi-hereditary rings, and got stuck with the following questions. Just to recall that being left semi-hereditary means that every finitely generated submodule of a ...
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0answers
116 views

Is a ring with stable range 2 2-Hermite?

Let $R$ be a (possibly non-commutative) ring. The left stable range of $R$ (denoted $sr_l(R)$) is the smallest $n$ such that every left unimodular row of length $>n$ is reducible. A similar ...
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1answer
826 views

If a quotient ring is a projective module then the ideal is principal

I've been studying projective modules in Rotman, as well as the topic of localization. Now on the Wikipedia article about them, there's an example of a locally free module that is not projective. The ...
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1answer
163 views

Morphisms in K-theory: comparison of two pictures

Algebraic $K_0$ group for an algebra $A$ may be defined in terms of stable isomorphism classes of idempotents in $M_n(A)$ or equivalently in terms of isomorphism clasess of finitely generated ...
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1answer
156 views

Projectives in the category of modular representations of Lie algebras

Let $\mathfrak{g}$ be a semi-simple Lie algebra (eg. $\mathfrak{g} = \mathfrak{sl}_n$), defined over an algebraically closed field $\textbf{k}$ with characteristic $p >> 0$. The center $Z(U\...
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1answer
123 views

Why is $\operatorname{nr}_{F[G]}:K_1(F[G])\to Z(F[G])^\times$ a bijection?

Let $A$ be a finite dimensional semisimple $F$-algebra and $K_1(A)$ the Whitehead group of $A$. By splitting $A$ into its Wedderburn components, the reduced norm map $\operatorname{nr}_A:K_1(A)\to Z(...
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1answer
236 views

Which varieties are flat degenerations of projective space?

Let $V$ be a vector space over a field with discrete valuation and let $R$ be its ring of integers. Which varieties can be reached as the special fiber of a flat degeneration over $R$, when the ...
3
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1answer
160 views

An invariant submodule of a projective module

This is a basic question (not research level) which has already been asked on SE by someone else but doesn't yet have an answer so I'd like to repost it on MO. Let $R$ be a commutative ring with ...
9
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1answer
351 views

Why do we want $p$-permutation modules in splendid equivalences?

First Rickard (in Splendid Equivalences: Derived Categories and Permutation Modules ) and then Rouquier (Block theory via stable and Rickard equivalences, Appendix A.1) define splendid equivalences ...
3
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1answer
281 views

A technical question about affine grassmanian

For a commutative ring $R$, consider $R[[t]]$-modules $$t^k R[[t]]^n \subset M \subset t^{-k} R[[t]]^n \subset R((t))^n.$$ It is known that if $t^{-k} R[[t]]^n / M$ is finitely generated projective $R$...
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0answers
72 views

Analogue of Bass's Lemma 2.4 on when inverse images of free modules are free

Let $R$ be a Noetherian integral domain. Let $x\in R$ be a prime element. Let $\overline{R}=R/Rx$. Let $P$ be a finitely-generated projective $R$-module. Assume that $\frac{P}{xP}$ is a free $\...
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0answers
147 views

Seeking an unpublished manuscript by Tetsuro Okuyama

Several papers in representation theory attribute the notion of relatively projective modules to Tetsuro Okuyama's manuscript "A generalization of projective covers of modules over finite group ...
7
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1answer
571 views

Vector bundles on open (affine) curves

It is well-known by Grothendieck (or earlier by Dedekind-Weber) that every vector bundle on $\mathbb{P}^1_k$ for $k$ a field decomposes into a sum of the line bundles $\mathcal{O}(k)$. As ...
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2answers
1k views

Every finitely generated flat module over a ring with finitely many minimal primes is projective

Over a commutative ring $R$, a finite type locally free (weak sense) module for which the rank function is locally constant is projective. If we notice that for each minimal prime $p$ of the ring, ...
8
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1answer
311 views

Jordan-Hölder-like statements for modules with $\Delta$-filtrations over a quasihereditary algebra

Definitions Let $A$ be an Artin algebra (for instance, take $A$ to be a finite dimensional algebra over some field) and label the isomorphism classes of simple $A$-modules by the elements of a ...
2
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1answer
178 views

Projective dimension of a quotient ring

Assume $A$ and $B$ are commutative algebras with $1$, $B = A[z] = A[Z]/(h(Z))$, $Z$ an indeterminate. The first comment in this question says that, if $A$ is noetherian, then $pd_{B\otimes_A B}(B) \...